The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

: A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a ﬂat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in ﬁelds such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature.


Introduction
The theory of curves and surfaces forms the basis of differential geometry. With developing technology and increasing studies, the theory of surfaces creates a very wide field of study. The theory of surfaces has many applications, especially in physics, engineering, design, and computer modeling. Thus, the geometric structure of special surfaces in Euclidean space has become an important field of study for geometricians [1][2][3][4]. According to the Frenet equations of the null curves in the semi-Euclidean 4-space, the conditions of existence and geometric characterizations of the Bertrand curves of the null curves were examined [5]. Some classical results of Bertrand curves for timelike ruled and developable surfaces were examined using an E-study map. In addition, some new results and theorems have been obtained regarding the developability of Bertrand offsets of timelike ruled surfaces [6].
The invention of the conchoid has been attributed to the Greek geometer Nicomedes by Pappus and other classical writers in the 2nd century BC. The conchoid is visually musselshell shaped. A conchoid is a type of curve derived from a fixed point, another curve, and a fixed length. Conchoids can be used in magnetic research, building construction, optics, physics, etc. It has been used in many applications such as [7][8][9]. Oruç et al. studied the planar and space conchoid curves and surfaces in three-dimensional Euclidean space [10,11]. Dede (2013) computed the types of spacelike conchoid curves in the Minkowski plane [12]. Aslan andŞekerci (2021) examined the condition which is the conchoidal surface and the surface of revolution given with a conchoid curve to be a Bonnet surface in Euclidean

Materials and Methods
Let E 3 = IR 3 , g defined by the metric x, y = x 1 y 1 + x 2 y 2 + x 3 y 3 be called Euclidean 3-space. Here, x = (x 1 , x 2 , x 3 ) and y = (y 1 , y 2 , y 3 ) are the standard coordinates of E 3 space. Let α : I → E 3 be a unit speed curve. The Serret-Frenet frame {T, N, B} of the curve α can be written by where ∧ is the vectoral product. The derivative formula of the Serret-Frenet frame in the matrix form can be calculated by equations where κ and τ are curvatures of the Serret-Frenet frame in Euclidean 3-space [3]. Let α : I → E 3 be a unit speed curve and α * : I → E 3 is given with the same interval. (α, M) and (α * , M) are the curve-surface pairs in E 3 and I ⊂ E 3 . An involute of a curve in Euclidean plane E 2 is a curve to which all tangent lines of the initial curve at corresponding points are orthogonal. If c = c(u) is a curve in E 2 parametrized by arc length, then the parametrizations of its involutes are where a ∈ IR is a constant, and t(u) = c (u). For spatial curves in Euclidean 3-space, the situation is more complex (see discussion in [2]). The most classical definition can be found by Eisenhart [1], where an involute is defined by the identity (1) for curves parametrized by arc length [1,2]. The pair (α, α * ) is said to be a Bertrand pair if their principal normal vector fields are linearly dependent at each point. If the curve α * is a Bertrand partner of α, then we may write that where λ is constant [3][4][5][6].
A conchoid curve α d (t) is a curve derived from a fixed point O, another curve α(t), and a constant length d. In this case, Q is the set of points on the line OP where there is a moving point P such that the distance between P and Q is d. For an analytic representation, it is convenient to choose O(0, 0). Using a representation of a curve α(t) in terms of the polar coordinates α(t) = r(t)(cos t, sin t), its conchoid α d (t) with respect to O and distance d is obtained as α d (t) = (r(t) ± d)(cos t, sin t). More generally, we can consider any parameterization k(t) of the unit circle S 1 . Then the curve α(t) and its conchoids α d (t) are represented by respectively, where k(t) = 1, refs. [7][8][9][10][11][12][13]. Let M be a smooth surface in E 3 given with the patch X(s, t) for (s, t) ∈ D ⊂ E 2 . The tangent plane to M at an arbitrary point P of M is spanned by X s (P) and X t (P), where the vector fields X s (P) and X t (P) denote derivatives with respect to s and t, respectively. The unit normal vector field of the surface M is where X s (s, t) ∧ X t (s, t) = √ EG − F 2 = W. The coefficients of the first and second fundamental forms on any T p M plane of the surface M are respectively The Gauss curvature K, mean curvature H of the surface M are calculated by respectively [3,4]. A surface of revolution is formed by revolving a plane curve about a line in E 3 . For an open interval I, let γ : I → Π be a curve in a plane Π and let l be a straight line in Π which does not intersect the curve γ on the Euclidean space E 3 . A rotation surface R is defined as a surface rotating the curve γ around l where they are called the profile curve and the axis, respectively. We may suppose that the axis l is the z -axis and the plane Π is the xz-plane, without loss of generality. Then the profile curve γ is given γ(u) = (u, 0, ϕ(u)). Hence a rotation surface R can be parametrized by Ref. [14]. In Euclidean 3-space, a twisted surface is obtained by rotating a plane curve about a line passing through its support plane, while the support plane is rotated about an axis. Let us assume that the profile curve α lies in the xz-plane. Thus, it can be parametrized as α(t) = ( f (t), 0, g(t)). If α is rotated about the straight line through the point (a, 0, 0) parallel to the y-axis, then we obtain using the anti-symmetric rotation matrix. By rotating (9) about the z-axis, the following twisted surface is obtained in Euclidean 3-space.
So, a parametrization of the twisted surface is defined by Note that since both rotations must be synchronized, they are expressed using the same parameter s. Here, the presence of the factor b ∈ R allows for differences in the rotation speed of both rotations [15,16]. The coefficients E, F, G and e, f , g of the first and second fundamental forms of the twisted surface X(s, t) in the Euclidean 3-space are and respectively [13][14][15]. Thus, from (7) the Gauss and mean curvature of the twisted surface X(s, t) in Euclidean 3-space are Let α be a planar curve given with profile α(t) = ( f (t), 0, g(t)) and α * be the involute of the α curve. In that case, the involute curve is written as The twisted surface formed by the involute of the curve α in Euclidean 3-space is given by ) sin(bs))(cos s, sin s, 0) Refs. [15][16][17][18]. Let α be a planar curve given with a profile curve α(t) = ( f (t), 0, g(t)) and β be the Bertrand curve pair of the α curve. In that case, the Bertrand curve is written as where λ is a real constant [18]. Thus, the twisted surface formed by the Bertrand pair of the curve α in Euclidean 3-space is cos(bs) . (22)

Corollary 1.
Let Ω(s, t) be a twisted surface given by (23). The coefficients E, F, G and e, f , g of the first and second fundamental forms of the twisted surface Ω(s, t) in Euclidean 3-space are and e(s, t) respectively.

Corollary 2.
Let Ω d (s, t) be a conchoidal twisted surface given by (24). The coefficients E, F, G and e, f , g of the first and second fundamental forms of the conchoidal twisted surface Ω d (s, t) in Euclidean 3-space are and respectively.
and where and Proof. Let α be a planar curve given with a profile curve α(t) = r(t)(cos t, 0, sin t) and α * in Equation (19) be the involute of the curve α.
Thus, a parametrization of the twisted surface formed by the involute of the curve α in (42) is obtained. Similarly, a parametrization of the conchoidal twisted surface formed by the involute of the curve α d is given.

Conclusions
In this study, some basic definitions were given for creating twisted and conchoidal twisted surfaces. The first and second fundamental forms of the conchoidal twisted surface were computed. Then, the Gaussian and mean curvature of the conchoidal twisted surface were calculated. Additionally, the conchoidal twisted surfaces formed by an involute curve and a Bertrand curve pair were defined.

Conclusions
In this study, some basic definitions were given for creating twisted and conchoidal twisted surfaces. The first and second fundamental forms of the conchoidal twisted surface were computed. Then, the Gaussian and mean curvature of the conchoidal twisted surface were calculated. Additionally, the conchoidal twisted surfaces formed by an involute curve and a Bertrand curve pair were defined.